NASA Technical Reports Server (NTRS) 19940006116: Harmonic oscillator interaction with squeezed radiation

N94-10571

HARMONIC OSCILLATOR INTERACTION
WITH SQUEEZED RADIATION

V. V- Dodonov, D - E- Nikonov
Moscow Institute of Physics and Technology
Zhukovsky, Moscow Region, 140160 Russia

Although the problem of the electromagnetic radiation by a
quantum harmonic oscillator is considered in textbooks on quantum
mechanics (see, e-g-, Ell) some its aspects seem to be not

clarified until now. By this we mean that usually the initial
quantum states of both the oscillator and the field are assumed
to be character i zed by a definite energy level of the oscillator
and definite occupation numbers of the field modes- In connection
with growing interest in squeezed states it would be interesting
to analize the general case when, the initial states of both
subsystems are arbitrary superpositions of energy eigenstates.
This problem was considered partly in Refs- 2—4, where the power

of the spontaneous emission was calculated in the case of an

arbitrary oscillator's initial state (but the field was supposed
to be initially in a vacuum state)- In the present article we
calculate the rate of the oscillator average energy and squeezing
and correlation parameter change under the influence of an

arbitrary external radiation field- Some other problems relating
to the interaction between quantum particles (atoms) or
oscillators with the electromagnetic radiation being in arbitrary
( in particular, squeezed) state were investigated, e-g- , in Refs
5-7-

Let us describe a charged harmonic
Hami 1 toni an

H = a* a

o

and the field by a hamiltonian

osci 1 1 ator

by a

( 1 )

61

ones of field

j

here u is the frequency of the oscillator, ax -
modes, a,b - correspondi ng destruction operators.

In a rather general case interaction can be described in a

form

H “hr fjja + b + + X a*b + H.c.l (3)

* j l J J J J J

(H.c. means hermitian conjugated part, /j and X. are constants).

In Schrodinger picture an arbitraty initial state vector
J^<0) > evolves into a state vector |v/ g (t)> as predicted by
Schrodinger equation with Hamiltonian H “ + H_ + H .

O It 4

In interaction picture any Schrodinger operator 0 changes

according to evolution operator U correspondi ng to H = H + H

O OR

Q(t> = U*(t)QU <t). (4)

o o

For example

a<t) “ a exp (-i cot), b.(t) = b. exp(-ioi.t). (5)

i j i

The interaction Hamiltonian in this picture

H “hr fu a*b + exp < i o> t + i o>t ) + X a + b exp (-i to t + i cot ) + H.c. 1 (6)

i j l J J J J } J J

generates evolution operator U<t) so that a state vector in this

picture defined as

IV'(t) > « U+(t> Iv^ft) > (7)

will variate according to

IV'(t) > “ U(t) | 0) >. <B>

Expectation value in this picture

<Q> j = <V'<t) |Q |v'<t) > (9)

variates slowly, only due to interaction. On the other hand, it is
related to the conventional expectation value as follows

<Q> “ <y/ |U OlT |w >. (10)

X S * O O 1 s

After introducing designations we can pose several questions
to answer:

1. Can absorption and emission be distinguished ijn a general case ?

2. Then how to calculate the rates of these processes ?

3- Is time ordering important in perturbation calculation for this
case ?

4. Does stimulated emission manifest itself ?

5- How does squeezing parameters of the oscillator and the field
vary 7

To calculate the rates of the processes we need to consider
infinitely long time intervals t + a in comparison with
oscillation period- But they must be much shorter than damping
time. Then the evolution operator has meaning of scattering matrix
S transforming initial state Jy^O) > s |i> to resulting one |r>.
From Heisenberg equation one gets

S = exp T <-iT) , (11)

where all products are believed time-ordered (designated with
subscript T) , and T - matrix is given by

co

T ■ / H <t)/h dt. (12)

i

-co

For our particular case

T = 2nh £ £ \.aV«5(«-w) + H.c-J, (13)

J

here the terms with p vanish because of a factor 6(ii>Tt*>) . Further

6 . = 6<w-a>) . Delta function originates as a limit of an integral
J J

T/ 2

Int ■ / exp(iOt) dt (14)

— T/ 2

(here the initial instant in time re-desi gnated as -t/2) •
Limits of this integral are

Int •* t , if ft -► 0, <14 ' )

Int -► 2 jt 6(0), if t + & (14")

Conventional techique in quantum electrodynamics is as
follows C81« T - matrix is splitted into two parts - absorption
part

T~ * 2tt r Va*b <5 (<*>.- u>) (15)

7 ) j j

and hermitian 'conjugated emission part T • Then probability for

63

( 16 )

time t of absorption < and similarly emission ) is declared as

P- - < i |T*T'|i> 5 I I < f l T "l i :> I*’

f

where summation is performed over a complete set of possible final
states- If rewritten in a form

T" - 2rrM'6(E f -E v ) , <17)

where E and E are energies of final and initial states, it shows

f i

employing (14) that (16) expresses the well-known Fermi's rule

2 nr _

P *» -T— Z \ <f l M |i > | • (18)

l

Put is it always valid and why probability is defined in this
manner ?

The expansion of S - matrix (11) i s as follows

S = 1 - i (T + +T*> - (T 2 ) /2 + ... (19)

T

The identity of normalisation must be valid in all orders of
perturbati on , i.e. for all powers of T as it is proportional to
the first power of coupling constant i

1 = < r |r> - < i | i > + <i |T + T + |i> + <i |T _ T* Ji >

+ <i |T*T~|i> + <i JT T |i> - <i j(T 2 > T |i> + (20)

Then terms from second to fifth can be interpreted as a
probability of transitions in the second order, since the first
and the sixth will be probability to stay in the initial state- So
conventional procedure ignores the second and the fifth terms- It
is possible only if T |i> is orthogonal to T |i>- It can happen
when either field or the oscillator is in energy eigenstate. Then
actually only two levels are involved in any sort of transitions-
In this case emission and absorption can be distinguished- That is
on obtaining after measurement one of Jf> states we can tell a
result of absorption from a result of emission-

For arbitrary initial state they cannot be distinguished
experimental ly- But the total probability of emission and
absorption together in (20) does not have physical meaning.
Therefore we have to revise our approach- More well-grounded

64

observabl e

procedure is to calculate not probabilities but
variations s

A<Q> m <i |S*QS|i> - < i |Q | i >. (21)

Besides, we do not need to introduce the Fock basis |f>, but deal
only with the initial state*

Since the observable variation is expected to grow with time,
to calculate the rates of the processes we need to consider only
terms proportional to long time t. Me will see later that

expressions like (21) contain terms with factor <5(to-<i>) and terms

2 )
with <5 (<j-ci>) under a sign of summation. One power of delta

function disappear because of summation over the continuum of

modes. The rest one power will transform to factor r. So terms

with delta function of infinitely little difference to the first

and zeroth powers will give non-growing with time observable

variation. Consequently, these terms represent dressing bare

states by virtual quanta. Terms with the second powers of delta

function will give time-proportional variations of observables.

Just these terms correspond

to transitions

with creation

of

real

quanta.

For our case we i

need 5

- matrix up to

the second

order

of

perturbati on . In this

order

a time-ordered

product

to

to

r

(T 2 ) - /

T -CO

dt /

i —to

dt H (t ) H (t

2^1 i I

)1 /h ,

2 JT 1

(22)

where

(h (t )H (t )

L- {

H (t )H (t ) ,

I 2 I 1 ’

if t > t ,

(23)

^ I 1 Z 2

JT l

H (t >H (t ) ,

I i I 2 *

if t > t ,
1 2 T

is different from non— ordered product

<T *>t = TxT * <24>

by a term

t

oo 2 r i

t .. .. * / dt / dt H (t )H (t > /h 2 , (25)

dtf -to 2 -to ill 2 X l J '

The latter expression depends on time like

*> 1 2 expCi (A-O) Z1 - 1

-to dt 2 -*> dt 4 ex P (i At , + i > ■ 4rr «5(A+0) lim . < 26 )

4 z+to i CA-Ci)

65

Such terms do not vanish only if A - -O. If A = u> + u. then the

last factor in <26’> is not singular. Terms with A * u>. - w or

2

opposite give a contribution to T ^

E l^l 2 Cab /*V

exp (2i AZ) -1+exp <-2i AZ>-1

which is not singular either. So T 2 ^ contains first powers of
delta functions and can be neglected compared to T (the former is
coupling constant X^ times less) •

We arrive to an assumption

S = 1 -iT - TxT/2, (28)

that leads to ,

A<Q> x - <i | i CT , QD - | CT, CT,Q33 |i> (29)

the first term being virtual and the second - real-

Strai ghtf orward calculation using (29) gives for example

A<a> - - E iX.2m5<b > - ^ E |A |*<2n6.) 2 <a> , (30a)

i T j J ) Z ' j' J

J *

A<b > = -iX*2n6 <a> - k X*2n«5 J] X 2m5<b >, <30b)

k I k k ic K . j j J

A<a*a> 85 i E <X*<b%t> - X.<a + b >> 2n6 - E |X . | 2 (2n6 > 2 <a*a>
i . J J J J J v k *

J

+ J E (X X*<b’b > + X*X <b + b >) <2n) z 6<5 . (30c)

2 “ . j k k j jkjk jk

k. J

A<b r b > = i (X <a + b > - X*<b*a>) 2m5, + |X | 2 <a a>(2m5 ,)

k k I k k kic k k

- n& v [ \ E X*2«A j <bJb k > + X* E X.2n<5.<bjV> J. <30d)

These variations are expressed in terms of expectation values in
the initial state (designated with triangle brackets). One can
define quadrature component variances by

(30a)

<30b )

(30c)

( 30d )

D(P,Q)

<pq> + <qp:

>]-

<PXQ>.

Their variations can be expressed similar to

AD(a f a)j ■ A<aa>j — 2<a> A<a> i — (A<a> j ) Z .

This kind of variance is important because in canonical

66

coordinate-momentum space ImD(a,a) corresponds to correlation and
ReD (a , a) - to squeezing. In Schrodinger picture they rapidly
transfer from each to other.

Retaining in (30) and (32) only terms proportional to t and
dividing by t we obtain time derivative equations* From them we
clearly see that radiation damping

r - Z |v|

determines variation of amplitudes

57 <a> x

_ t
2

ln6 .

<a>,

— <b, > - - = |\J 2 2n6 <b.>.

dt k I 2 1 k > k k

(33)

<34a)

(34b)

These equations
frame of Wigner

coincide with those obtained usually in the
- Weisskopf approximation. Field modes and the
oscillator exchange their energies. As a result there is no effect
of stimulated emission but only two independent fluxes of energy:

d

dt

k k I

■ - y<a*a> + £
k

|\ | 2 2n6 k <a 1 'a>

IM z 2"V b k b k>’

+ l\l 22 ^ k < b k b k > -

(34c)

<34d)

Spueez i ng-correl at i on
d

— D(a,a) i -

d

dt

D(b,

k

b, ) = -

k I

parameter behaves in a similar way :
- yD(a,a) - £ X* 2n6 k D <b k ,b k > ,

k

\* 2 2rr t 5 k D(a,a) - \\ | *21x6,0 <b k ,b fc > .

(34e)

(34f )

Further development can be made for the specific expressions
of coefficients in Hamiltonian (3). For the continuum of modes
summation is substituted by integration over phase space and
summation over polarization indexes r

with volume V,

£ -> £ / Vp do> dfi

j *

solid angle element dO, mode frequency

2

Q>

_ 9 3

Br: c

(35)

densi ty

(36)

67

Decomposition of vector potential A<r,t) over mode variables is

A ( r- , t )

E

j

j °

e. <b (t)exp(ik.r) + H.c->-

j

j

( 37 )

j-th

where e is a polarization vector and k. - a wave vector of

j }

mode*

Gauge invariance substitution of oscillator momentum p -» p -
eA leads to the interaction Hamiltonian (3)

-epA e 2 A z

H = — — + -r- - (38)

4 m 2fn

Here are the charge and the mass of the oscillator* The second

term in this case proves to be a unity operator in state-space of
the oscillator. Hence it results in an infinitely little
renormal i z at i on of field energy because of a factor 1/V (for
infinitely large volume V). The coupling constant will be

X

J

-‘if

o >

<i> m^ V
i °

COS© I
J

(39)

where ©. is the angle between a polarization vector and the
oscillation direction.

On the other hand, from the Hamiltonian in another gauge form

H ■ - eqE, (40)

where q is a coordinate of the oscillator and E is the electric
field vector, it follows that the coupling constant

X' » X. - j . (41)

j j u>

But as all expressions contain delta functions 6<u>-^) , constants

(39) and (41) coincide- We see that it is one of the cases when
gauge transform, performed over state vectors in the absence of
vector potential and correspondi ng to a change from gauge form

(40) to (38) , does not make any difference. These transforms were
considered in detail in Ref. 10.

Einstein’s stimulated coefficient can be also introduced.
However it is different from a common one - it depends on the
angle and expresses radiation power instead of probability i

68

(42)

B = 2nV | X. | 2

2 2 ^

ne cos 6

2m e

The spontaneous emission coefficient is obtained from (33).
Integration should be performed over solid angles of polarisation
vectors (they are also isotropically distributed), not wave
vectors of modes s

2 2

> <JL>

f B dfi = 7 9

. a a J 6rrm£ c

4n c o

(43)

A light beam containing several close modes has an energy density

W - r pCb + b>ho>

u> *“

r

or

W - / W do.

u>

It will allow us to

express eqs- (34) through

meaningful 1 values.

d r -s

dtH* *>]

■ - ^hoj<a + a> + / BW dfi,

Cl)

*M

« B ^phc*>< a *a > - wj ,

(44)

(45)

physi cal 1 y

(46a)

(46b)

d

dt

D (a , a) ■

d

dt

[wVD(b,b) j

( a , a ) + / BW

D <b,b)

" hco<b + b>

dO,

B £phoj<b*b>D (a, a) - W^D(b,b)j.

< 46c )
(46d )

All above discussed enables us to answer posed questions i
1* In general absorption and emission can not be distinguished.

2- So not Fermi's rule but expectation values should be used to
calculate the rates of these processes*

3. Time ordering in this case is not important up to the second
order of perturbati on •

4* Stimulated emission does not manifest itself in the final
resul t .

5* Energy and squeez i ng-correl ation parameters behave in a similar
way : there are independent interchange fldxes of them

proportional to their current values-

69

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Rad i ophysi cs 1 (Moscow, Sovetskoye Radio, 1965).

3. E- Santos, The Harmonic Oscillator In Stochastic
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70